\[
  \underline{\underline{\text{Nérm}}}_{A/S} \longrightarrow
  \underline{\underline{\mathrm{Hom}}}_{S\text{-}\mathrm{gr}}(A, A^{*})
\]

\[
  \boxed{\ \pi^{*}(\xi) - \mathrm{pr}_{1}(\xi) - \mathrm{pr}_{2}(\xi)\ }
\]

\[
  \dim A \;\leq\; \dim \mathrm{Pic}\,A \;\leq\;
  \dim H^{1}(A, \mathcal{O}_{A}) \;\leq\; \dim A
\]

\[
  \alpha \rightsquigarrow \alpha' = \struck{\ill{}}\ \varphi_{\xi}^{-1}\,
  {}^{t}\alpha\,\varphi_{\xi}
\]

\[
  \mathrm{tr}(\alpha\beta') = \frac{r}{I(\xi, \ldots, \xi)}\,
  I(\xi, \ldots, \xi, D_{\xi}(\alpha\beta'))
\]

\[
  \mathrm{tr}\,\alpha\alpha' = \frac{r}{I(\xi, \ldots, \xi)}\,
  I(\xi, \ldots, \xi, 2\alpha^{*}(\xi))
\]

\[
  (\xi_{1}, \ldots, \xi_{r}) \rightsquigarrow I(\xi_{1}, \ldots, \xi_{r})
  \in \struck{\mathbb{Q}}\ \mathbb{Z}
\]

\[
  I(u^{*}(\xi_{1}), \ldots, u^{*}(\xi_{r})) = \nu(u)\,
  I(\xi_{1}, \ldots, \xi_{r})
\]

\[
  \nu(\alpha) = \frac{1}{I(\xi, \ldots, \xi)}\,
  I(\alpha^{*}(\xi), \ldots, \alpha^{*}(\xi))
\]

\[
  \nu(\alpha + n\cdot 1_{A}) = n^{2r} + \sigma_{1}(\alpha)\,n^{2r-1} +
  \cdots + \sigma_{2r}(\alpha)
\]

\[
  \mathrm{tr}(\alpha) = \frac{r}{I(\xi, \ldots, \xi)}\,
  I(\xi, \ldots, \xi, D(\alpha))
\]

\[
  N^{+}(A),\quad N^{>}(A) \;\subset\; N(A)
\]

\[
  N(A\times B) \big/ N(A)\times N(B) \;\xrightarrow{\ \sim\ }\;
  \mathrm{Hom}(A, D(B))
\]

\[
  s_{\xi}(\alpha) = \varphi_{\xi}^{-1}\,{}^{t}\alpha\,\varphi_{\xi}
\]

\[
  \text{(1)}\qquad D_{\xi}(\alpha,\beta) = (\alpha+\beta)^{*}(\xi) -
  \alpha^{*}(\xi) - \beta^{*}(\xi) \in N(A)
\]

\[
  \varphi_{D_{\xi}(\alpha,\beta)} = {}^{t}(\alpha+\beta)\,\varphi_{\xi}\,
  (\alpha+\beta) - {}^{t}\alpha\,\varphi_{\xi}\,\alpha -
  {}^{t}\beta\,\varphi_{\xi}\,\beta
\]

\[
  \text{(1$'$)}\qquad \varphi_{D_{\xi}(\alpha,\beta)} =
  {}^{t}\beta\,\varphi_{\xi}\,\alpha + {}^{t}\alpha\,\varphi_{\xi}\,\beta
  = \varphi_{\xi}(\alpha'\beta + \beta'\alpha)
\]

\[
  \text{(2)}\qquad D_{\xi}(\alpha) = D_{\xi}(\alpha, \mathrm{id}_{A}) =
  \varphi_{\xi}(\alpha' + \alpha)
\]

\[
  \text{(3)}\qquad D_{\xi}(\alpha,\beta) = D_{\xi}(\alpha'\beta) =
  D_{\xi}(\beta'\alpha), \qquad D_{\xi}(\alpha) = D_{\xi}(\alpha')
\]

\[
  B(A,B) = \mathrm{Hom}(A, D(B)) \;\xrightarrow[\ \simeq\ ]{s}\;
  \mathrm{Hom}(B, D(A))
\]

\[
  \boxed{\ N(A) \subset B(A,A)\ }
\]

\[
  \varphi_{\xi} : A\times B \longrightarrow D(A)\times D(B)
\]

\[
  \begin{pmatrix}
    \alpha_{\xi} & {}^{t}\lambda_{\xi} \\
    -\lambda_{\xi} & \beta_{\xi}
  \end{pmatrix},
  \qquad \lambda_{\xi} \in \mathrm{Hom}(A, D(B)) = B(A,B),\quad
  \alpha_{\xi} \in B(A,A),\quad \beta_{\xi} \in B(B,B)
\]

\[
  \boxed{\ \xi \longmapsto \lambda_{\xi} : N(A\times B) \longrightarrow
  \mathrm{Hom}(A, D(B)) = B(A,B)\ }
\]